Mathematics

We determine a positive real number (weight), which corresponds to a vertex of a tetrahedron and it depends on the three weights which correspond to the other three vertices and an infinitesimal number ϵ. As a limiting case, for ϵ→0, the quad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with the three neighbouring edges of the tetrahedron and intersect at this vertex.

In this paper, we introduce and investigate the subclass P ξ,κ p,q (τ,η) of starlike functions with negative coefficients by using the differential operator Υ ξ,κ τ,p,q . Coefficient inequalities, growth and distortion theorems, closure theorems, and some properties of several functions belonging to this class are obtained. We also determine the radii of close-to-convexity, starlikeness, and convexity for functions belonging to the class P ξ,κ p,q (τ,η) . Furthermore, we obtain the integral means inequality and neighborhood results for functions belonging to the class P ξ,κ p,q (τ,η) . The results presented in this paper generalize or improve those in related works of several earlier authors.

A new inequality, (x ) p +(1?�x ) 1 p ?? for p?? and 1 2 ?�x?? is found and proved. The inequality looks elegant as it integrates two number pairs ( x and 1?�x , p and 1 p ) whose summation and product are one. Its right hand side, 1 , is the strict upper bound of the left hand side. The equality cannot be categorized into any known type of inequalities such as Hölder, Minkowski etc. In proving it, transcendental equations have been met with, so some novel techniques have been built to get over the difficulty.

It's well known that every prime number p≥5 has the form 6k−1 or 6k+1 . We'll call k the generator of p . Twin primes are distinghuished due to a common generator for each pair. Therefore it makes sense to search for the Twin Primes on the level of their generators. This paper present a new approach to prove the Twin Prime Conjecture by a sieve method to extract all Twin Primes on the level of the Twin Prime Generators. We define the ω p n --numbers x as numbers for which holds that 6x−1 and 6x+1 are coprime to the prime p n . By dint of the average distance δ ¯ ( p n ) between the ω p n --numbers we can prove the Twin Prime Conjecture indirectly.

This paper discusses a few main topics in Number Theory, such as the Möbius function and its generalization, leading up to the derivation of neat power series for the prime counting function, π(x) , and the prime-power counting function, J(x) . Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given F a (s) , we know a(n) , which implies the Riemann hypothesis, and enabled the creation of a formula for π(x) in the first place), and the realization that sums of divisors and the Möbius function are particular cases of a more general concept. From this result, one concludes that it's not necessary to resort to the zeros of the analytic continuation of the zeta function to obtain π(x) .

This article proves a Pythagoras-type formula for the sides and diagonals of a polygon inscribed in a semicircle having one of the sides of the polygon as diameter.

The analysis and visualization of tensor fields is a very challenging task. Besides the cases of zeroth- and first-order tensors, most techniques focus on symmetric second-order tensors. Only a few works concern totally symmetric tensors of higher-order. Work on other tensors of higher-order than two is exceptionally rare. We believe that one major reason for this gap is the lack of knowledge about suitable tensor decompositions for the general higher-order tensors. We focus here on three dimensions as most applications are concerned with three-dimensional space. A lot of work on symmetric second-order tensors uses the spectral decomposition. The work on totally symmetric higher-order tensors deals frequently with a decomposition based on spherical harmonics. These decompositions do not directly apply to general tensors of higher-order in three dimensions. However, another option available is the deviatoric decomposition for such tensors, splitting them into deviators. Together with the multipole representation of deviators, it allows to describe any tensor in three dimensions uniquely by a set of directions and non-negative scalars. The specific appeal of this methodology is its general applicability, opening up a potentially general route to tensor interpretation. The underlying concepts, however, are not broadly understood in the engineering community. In this article, we therefore gather information about this decomposition from a range of literature sources. The goal is to collect and prepare the material for further analysis and give other researchers the chance to work in this direction. This article wants to stimulate the use of this decomposition and the search for interpretation of this unique algebraic property. A first step in this direction is given by a detailed analysis of the multipole representation of symmetric second-order three-dimensional tensors.

Let n≥2 and K be a number field of characteristic 0 . Jacobian Conjecture asserts for a polynomial map P from K n to itself, if the determinant of its Jacobian matrix is a nonzero constant in K then the inverse P −1 exists and is also a polynomial map. This conjecture was firstly proposed by Keller in 1939 for K n = C 2 and put in Smale's 1998 list of Mathematical Problems for the Next Century. This study is going to present a proof for the conjecture. Our proof is based on Dru{ż}kowski Map and Hadamard's Diffeomorphism Theorem, and additionally uses some optimization idea.

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial. Tonelli-Shanks algorithm remains the most widely used and probably the fastest when averaged over all primes [19]. This paper proposes a new algorithm for finding square roots modulo all odd primes, which shows improvement over existing method in practical terms although asymptotically gives the same run time as Tonelli-Shanks. Apart from practically efficient computation time, the proposed method does not necessarily require availability of non-residue and can work with `relative non-residue' also. Such `relative non-residues' are much easier to find ( probability 2/3) compared to non-residues ( probability 1/2).

We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.